Download 3. Right Triangle Trigonometry

3. Right Triangle Trigonometry
3.1
3.2
3.3
3.4
3.5
Reference Angle
Radians and Degrees
Definition III: Circular Functions
Arc Length and Area of a Sector
Velocities
1
3.1 Reference Angle
•
•
•
•
Reference Angle
Reference angle theorem
Calculate trig function using reference angle
Approximation
2
3.1 Reference Angle
•
Definition. The reference angle for any angle θ
in standard position is the positive acute angle
between terminal side of θ and the x-axis. We
denote the reference angle for θ by θˆ
ˆ
θ θ =θ
θ
θˆ
θˆ
θ
θ
θˆ
3
3.1 Reference Angle
Reference Angle Theorem
A trigonometric function of an angle and its reference
angle differ at most in sign.
You need to remember the signs of trig function on
each quadrant (Section 1.3, table 1)!
4
3.1 Reference Angle
Problems.
(1) Draw the given angle in standard position and name the
reference angle.
(a) 253.8° [4]
Ans. 73.8°
(b)
–330° [10]
Ans. 30°
(2) Find the exact value
(a) cos135° [14]
−
2
2
(b) csc300°
−
[24]
2
3
5
3.1 Reference Angle
Problems.
(3) Use calculate to find
(a) cos238°
[30]
Ans. –cos(58°) = –0.5299
(b)
csc93.2°
[36]
Ans. csc(3.2°) = 1.0016
(4) Find θ to the nearest tenth of degree, 0° < θ < 360°,
and if
[50, 62]
(a) sinθ = –0.3090 and θ ∈ QIV; (b) secθ = –3.4159 and θ ∈ QII
Ans. 342.0°
Ans. 107.0°
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3.1 Reference Angle
Problems.
(5) Find θ, 0° < θ < 360°, and if
•
sinθ = −
•
secθ = 2 and ∈ QIIV
1
2
and θ ∈ QIII
[68]
Ans. 225°
[78]
Ans. 315°
7
3.2 Radians and Degrees
•
•
•
Radian measure of an angle
Conversion between radians and degrees
Using calculator
8
3.2 Radians and Degrees
•
Radian measure of an angle
A
O
θ
s
O: center of circle
θ : central angle
r: radius of circle
AB: arc
s: is the arc length
r
B
s
θ (in radians) =
r
In particular, if s = r, then we get an central angle with
exactly one radian.
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3.2 Radians and Degrees
Radian and degree
•
⎛ 180 ⎞
1 (rad) = ⎜
⎟
⎝ π ⎠
•
1 =
o
π
180
(rad)
o
where π ≈ 3.1415926
Multiply by
Degrees
Multiply by
π
180
Radians
180
π
10
3.2 Radians and Degrees
problems
(1) Let θ be a central angle in a circle of radius r with arc
length s. Find θ if [2, 6]
(a) r = 10 in, s = 5 in
Ans. 0.5 radians
(b) r = 14 cm, s =
1
8
cm
Ans. 0.5 radians
(2) For each given angle, [14, 16]
• Draw the angle in standard position
• Convert the angle to radian measure
• Name the reference angle to both degrees and radians
(a) –120°
(b) 390°
Ans. ref = 60°, π/3 (rad)
Ans. Ref = 30°, π/6 (rad)
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3.2 Radians and Degrees
problems
(3) Give the exact value of each of the following.
(a) cot π3
[56]
(b) sec 56π
1
3
[60]
2
3
(4) For each given angle, [44, 46]
• Draw the angle in standard position
• Convert the angle to degree measure
• Name the reference angle to both degrees and radians
(a) 116π
(b) − 53π
Ans. ref = 30°, π/6 (rad)
Ans. Ref = 60°, π/3 (rad)
12
3.2 Radians and Degrees
problems
(5) Evaluate each expression using x = π6 . Provide exact
answer.
[74, 78]
(a) sin (x − π3 )
− 12
(b) − 1 + 43 cos(2 x − π2 )
−1+ 3 3
8
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3.3 Definition III: Circular Functions
•
•
•
•
3rd definition of trig functions
Values of trig function at special angles
Domain and range of trig functions
Problems
14
3.3 Definition III: Circular Functions
If (x, y) is any point on the unit circle, and t is the distance from
(1, 0) along the circumference of the unit circle, then the six
Trigonometric Functions are defined as below.
Trig Function
(circular function)
sin t = y
cos t = x
y
tan t = x
cot t = xy
sec t = 1x
csc t = 1y
y
(x, y)
t
θ =t
(1, 0)
x
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3.3 Definition III: Circular Functions
(−
(−
3
2
,
1
2
1
2
,
(−
)
1
2
,
3
2
2π
3
3
2
1
2
π
2
120°
135° 3π
)
)
(, )
(
(0, 1)
90°
60°
π
3
π
4
4
1 150° 5π
6
2
1
2
45°
π
6
180° (–1,0) π
(−
(−
3
2
,− 12
1
2
,−
)
30°
(
)
3
2
, 12
)
0°
2π 360° (1,0)
7π
6
210°
1
2
1
2
,
)
(−
11π
6
5π
4
4π
3
225°
240°
1
2
,−
3
2
)
3π
2
270°
(0, –1)
5π
3
7π
4
1
2
3
2
(
(
3
2
,− 12
1
2
)
315°
( ,− )
300°
330°
1
2
,−
)
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3.3 Definition III: Circular Functions
Domain and range of trig functions
Domain of the circular functions (trig function)
sin t, cos t
all real numbers
tan t, sec t
all real numbers except t = π/2 + kπ for any integer k
cot t, csc t
all real numbers except t = kπ for any integer k
Range of the circular functions (trig function)
sin t, cos t
[–1, 1]
tan t, sec t
all real numbers, or (–∞, ∞)
cot t, csc t
(–∞, –1]∪[1, ∞), or all real numbers except (–1, 1)
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3.3 Definition III: Circular Functions
Problems
(1) Use the unit circle to evaluate each function.
(a) cos(225°)
[2]
(b) cot ( 43π )
−
[12]
1
3
2
2
(2) Using the unit circle, find the six trigonometric
function for θ = 53π .
[16]
sin(θ) = −
csc(θ) = −
3
2
2
3
cos(θ) =
1
2
sec(θ) = 2
tan(θ) = − 3
cot(θ) = −
1
3
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3.3 Definition III: Circular Functions
Problems
(3) Using the unit circle, find all θ between 0 and 2π and
for which sin θ = − 12.
[22]
7π
11π
and
6
6
(4) If angle θ is in standard position and the terminal side
of θ intersects the unit circle at the point.
[44]
(–1/ 10 ,–3/ 10 ), find sinθ, cosθ, and tanθ.
sin(θ ) = −
3
10
, cos(θ ) = −
1
10
and tanθ = 3
19
3.3 Definition III: Circular Functions
Problems
(5) Identify the argument of each function. [54, 58]
(a) sin(2A)
(b) cos(2 x − 32π )
ans. 2 x − 32π
Ans. 2A
(6) Use circular function definition to evaluate, [68, 70]
(a) cos(2π + π2 )
(b) cos(2π + π3 )
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3.3 Definition III: Circular Functions
Problems (related to domain and range)
(7) Determine if the statement is possible for some real
number z ?
[74, 76, 80]
(a) csc 0 = z
(b) sin 0 = z
(c) sin z = 1.2
Ans.
No
Ans.
Yes
Ans.
No
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3.4 Applications
1) Formula for Arc Length (in a circle)
s = rθ
r
θ must be in radians
s
θ
2) Formula for Area of a Sector
A = 12 r 2θ
Again, θ must be in radians
r
θ
s
22
3.4 Applications
(1) Find the arc length s of a circle with central angle θ =
30° and radius r = 4 mm. [8]
2π
3
mm
(2) Find the radius r of a circle with central angle θ =
and arc length s = π cm.
[34]
4
3
3π
4
cm
(3) The minute hand of a clock is 1.2 cm long. How far
does the tip of the minute hand travel in 40 minutes?
[12]
1.6π cm
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3.4 Applications
(4) Find the area of the sector formed by central angle θ =
15° and r = 10 m.
[46]
25π
6
m2
(5) Example 6. If a sector formed by a central angle of 15°
has an area of π/3 square centimeters, find the radius
of the circle.
2 2 cm
24
3.5 Velocities
r
Uniform Circular Motion
s
Linear velocity: ν =
t
Angular velocity: ω =
θ
θ
s
θ in radians
t
Relation between two velocities: ν = rω
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3.5 Velocities
(1) Find the linear velocity ν of a point moving with
uniform circular motion if the point covers a distance s
= 12 cm in time t = 2 seconds.
[4]
ν = 6 cm/sec
(2) Find the distance s covered by a point moving with
linear velocity ν = 10 feet per second for a time = 4
seconds.
[8]
s = 40 ft
26
3.5 Velocities
(3) Find the distance s traveled by a point with angular
velocity ω = 2 radians per second on a circle of radius
r = 4 inches for time t = 5 seconds.
[24]
s = 40 in
(4) Find the angular velocity ω associated with 20 rpm.
[30]
ω = 40 rad/min
27